Linearized Riemann solvers are based on the exact solution of
Riemann problems of a modified system of conservation equations
obtained by a suitable linearization of the original system. This
idea was put forward by Roe [155], who developed a linearized Riemann solver for the equations of
ideal (classical) gas dynamics. Eulderink at al. [49,
50] have extended Roe's Riemann solver to the general relativistic
system of equations in arbitrary spacetimes. Eulderink uses a
local linearization of the Jacobian matrices of the system
fulfilling the properties demanded by Roe in his original paper.

Let
be the Jacobian matrix associated with one of the fluxes
of the original system, and
the vector of unknowns. Then, the locally constant matrix
, depending on
and
(the left and right state defining the local Riemann problem)
must have the following four properties:

It constitutes a linear mapping from the vector space
to the vector space
.

As
.

For any
,
.

The eigenvectors of
are linearly independent.

Conditions 1 and 2 are necessary if one is to recover smoothly
the linearized algorithm from the nonlinear version. Condition 3
(supposing 4 is fulfilled) ensures that if a single discontinuity
is located at the interface, then the solution of the linearized
problem is the exact solution of the nonlinear Riemann problem.

Once a matrix
satisfying Roe's conditions has been obtained for every
numerical interface, the numerical fluxes are computed by solving
the locally linear system. Roe's numerical flux is then given
by

with

where
,
, and
are the eigenvalues and the right and left eigenvectors of
, respectively (p
runs from 1 to the number of equations of the system).

Roe's linearization for the relativistic system of equations
in a general spacetime can be expressed in terms of the average
state [49,
50]

with

and

where
g
is the determinant of the metric tensor
. The role played by the density
in case of the Cartesian non-relativistic Roe solver as a weight
for averaging, is taken over in the relativistic variant by
k, which apart from geometrical factors tends to
in the non-relativistic limit. A Riemann solver for special
relativistic flows and the generalization of Roe's solver to the
Euler equations in arbitrary coordinate systems are easily
deduced from Eulderink's work. The results obtained in 1D test
problems for ultra-relativistic flows (up to Lorentz factors 625)
in the presence of strong discontinuities and large gravitational
background fields demonstrate the excellent performance of the
Eulderink-Roe solver [50].

Relaxing condition 3 above, Roe's solver is no longer exact
for shocks but still produces accurate solutions, and moreover,
the remaining conditions are fulfilled by a large number of
averages. The 1D general relativistic hydrodynamic code developed
by Romero et al. [157] uses flux formula (26) with an arithmetic average of the primitive variables at both
sides of the interface. It has successfully passed a long series
of tests including the spherical version of the relativistic
shock reflection (see Section
6.1).

Roe's original idea has been exploited in the so-called local
characteristic approach (see, e.g., [198]). This approach relies on a local linearization of the system
of equations by defining at each point a set of characteristic
variables, which obey a system of uncoupled scalar equations.
This approach has proven to be very successful, because it allows
for the extension to systems of scalar nonlinear methods. Based
on the local characteristic approach are the methods developed by
Marquina et al. [106] and Dolezal & Wong [42], which both use high-order reconstructions of the numerical
characteristic fluxes, namely PHM [106] and ENO [42] (see Section
9.4).